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The Quotient Category of a Morita Context

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The Quotient Category of a Morita Context CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector JOURNAL OF ALGEBXA 28, 389-407 (1974) The Quotient Category of a Morita Context BRUNO J. MUELLER* Department of Mathematics, McMaster University, Hamilton, Ontario, Canada Communicated by P. M. Cohn Received July 31, 1972 Every Morita context between rings R and S leads to an eq.uivalence between two quotient categories of the module categories mod R and mod S. As consequences, one obtains a generalization of the Morita Theorems, and one constructs induced contexts between quotient rings of R and S. The concept of context-equivalence of rings is introduced and se&died. The last part reviews and reorganizes various topics utilizing the new notions and results. 1. PRELIMINARIES DEFINITION 1.1. A (Morita) context consists of two rings R and S, two bimodules sPR and RQS, and two bimodule homomorphisms (called the pairings) (-, -): Q OS P -+ R and [-, -1: P OR Q + S satisfying the associativity conditions ~[p, 2’1 = (9, p) 4’ and p(q, p’) = [p, 41 p’. The inrages of the pairings are called the trace ideals of the context, and are denoted by TR and T, . Remarks. This concept was introduced in [4], and under the name of preequivalence data in [5, II. 3.21 (cf. also [6]). It is extensively used in [I], and appears in disguise in many investigations. The most elegant (but apparently useless) description of a Morita context is to say that it is just an additive category with two objects. Another description is obtained by constructing the matrix ring fl = (: $) and its idempotent e = (i z): a context is just a ring together with an idempotent. We abbreviate a context by the symbol <P, Q). From any module Pn and ring isomorphism C: S - endo P, one constructs a context by putting Q = hom,(P, R), (M, p) = a(p) and [p, m] = u-r(p~~); this context will be denoted by <P, 0) and called the derived context of P and O. If S = endo PR and D is the identity, one says simply derived contexi of P (cf. [6, p. 441; * The author acknowledges support by the National Research Council of Canada, Grant A-4033. 389 CopyTight 0 1974 by Academic Press, Inc. All rights of reproducdon in any form reserved. &T/28/7-2 390 BRUNO J. MiiLLER [5, 1.4.21). The trace TR of the derived context of PE is also called the trace of PR , trace(P,). We need a somewhat more restricted notion of isomorphism of contexts than the one suggested by the description as additive categories, namely the following: Two contexts are isomorphic, (P, Q) g (P’, Q’>, if they involve the same pair of rings R and S, and if there exist bimodule iso- morphismsf: P + P’ and g: Q -+ Q2’ which are compatible with the pairings. A context will be called a subcontext of another one, (P, Q) < (P’, Q’), if they satisfy the same conditions but with f and g only monomorphisms. Thus, every subcontext is isomorphic to a context whose bimodules are actually submodules and whose pairings are obtained by restriction. EXAMPLES 1.2. The only general method of constructing contexts is that of forming the derived context of a module; unfortunately it conceals the symmetry inherent in the original definition. A slight generalization restores this symmetry: From two modules X, and YA , define R = endo X, , S = endo YA , P = hom,(X, Y) and Q = hom,(Y, X), with pairings by composition. A context satisfies Ts = S iff PR and sQ are finitely generated projective, ,P and Qs are generators and the natural maps Q + P*, P -+ *Q, S + endo PR and S -+ endo aQ are bijective; iff PR is finitely generated projective, .P faithful and Q -+ P* surjective; iff ,P is a generator and Q- *P surjective. Then, TR is idempotent, TRQ = Q and PT, = P, and the context is isomorphic to (PR , CT) and (aQ, a), for the obvious choices of a (cf., [5, 11.3.41). A context satisfies T, = S and TR = R iff PR is a finitely generated projective generator and S-+endoP, is bijective (cf. [5,11.3.5; also Sect. 2.31). We finally mention the identity context (R, R> with pairings by multi- plication, and the trivial context <P, Q> with zero pairings. 1.3. This work was inspired by papers of Kato [26, 271 who proves our basic Theorem 3 in a slightly different language, for derived contexts. A great portion of our results was developed independently by Cunningham, Rutter and Turnidge [9] for the case of a derived context of a finitely generated projective module. The vast majority of related papers are concerned explicitly or implicitly with the derived context of a finitely generated projective module PR [2, 3, 7, 8, 9, 12, 15, 26, 34, 35, 381 or a generator .P [23, 40, 421, the notable exception being [l]. The first case yields exactly the contexts (P, Q,> with Ts = S discussed before, and the second one the slightly more special contexts for which Pa satisfies in addition the bicom- mutator relation. One obtains a number of simplifications, mainly due to the functor equivalence hom,(P*, -) E P @a - and the idempotency of TR . THE QUOTIENT CATEGORY OF A MORITA CONTEXT 391 1.4. All rings are assumed to have identity elements, all modules are unitary, and mod R denotes the category of all right R-modules. X* is used for the dual hom,(X, R) of a right R-module X, , similarly *Y for a left R-module RY. dim X denotes the Goldie dimension of the module X, i.e., the largest number of nonzero summands in a direct sum of submodules. The symbols 5, 5, rad, ‘I), ‘$I and * will be used consistently for the torsion free class, torsion class, torsion radical, Gabriel filter (additive topology in [39, p. 12]), quotient category (Giraud subcategory, full subcategory of L&closed modules in [39, pp. 47-48]) and quotient functor of any hereditary torsion theory on a module category. Stenstriim [39] may serve as a general reference on hereditary torsion theories. 2. THE QUOTIENT CATEGORY 2.1. We start by collecting information about the hereditary torsion theory on mod R determined by an ideal T of R. PROPOSITION 1. For any two-sided ideal T of a ring R, is the torsionfree class of a hereditary torsion theory. The corresponding Gabriel jilter is the set 3 of right ideals I such that ann,T = 0 implies annxr = 0 for all X E mod R, and the corresponding quotient category 58 consists of the X E mod R for which the natural X -+ hom,(T, X) is bijective. Proof. One verifies without difficulty that 5 is closed under products, submodules,extensions and essentialmonomorphisms, hence is a torsionfree class.Then I E D iff R/I is torsion, i.e., 0 = hom,(R/I, X) - ann,T for all. X E 5, providing the description of 3. The natural map X 33~ti (t +-+xt) E hom,(T, X) is injective iff ann,T = 0, i.e., XE 5. Under this assumption hom,(T, X) may be identified with the set of those elementse of the injective hull E of X which satisfy eT C X. Therefore the natural map is surjective iff eT C X implies e E X, i.e., annEfxT L- 0 or E/X E &. But for any hereditary torsion theory, X E 9l ifLFX, E/X E 3, COROLLARY 2. The Gabriel Jilter consists of all right ideals containing T, a2 T = T2. Proof. Clearly T LL-:T” is necessaryfor the set of right ideals containing 392 BRCNQ J. Mi:LLER T to be an idempotent filter. If this condition holds, consider the module X = R/(IT : T) for any IE 3. We have ann,T = 0 hence ann,I = 0, but iI = 0 hence icann,I = 0; i.e., 1 E(IT: T) or TCITCI. Remark. One would like to know which Grothendieck categories occur as quotient categories determined by ideals. The case of idempotent ideals is settled in [36]. ‘I’HEORRM 3. Every Morita context <P, Q> betweenrings R and S indzcces an equivalencebetween the quotient categories2& and 21sof mod R and mod S determinedby the two trace ideals. Proof. The bimodules sPR and RQ!S determine functors hom,(P, -): mod R + mod S and hom,(Q, -): mod S + mod R, and the pairings induce natural transformations 4: &ocm -j hom,(Q $31~ P, -) z homs(Q, hom,(P, -)> and ‘Pz i&-m -+ homR(P, hom3(Q, -I>, explicitely given by &(x)(q)(p) =: x(q, p). We claim $r to be isomorphic exactly if X E ‘LI, . Clearly ker & = ann,T where T = (Q, P) is the trace ideal; hence +x is injective iff X E 3. Under this assumption, any map 01 E hom,(Q OS P, X) factors over ( , ),: Q OS I’ + T, since C (qi , pi) = 0 implies a(C 4i 0 A)(% PI, = aE 4i 0 Pi(%P,) = a(1 Qi0 [Pi Y4lP) = a(C 4ii3i >41 OP) = “ICY& 1Pi)4 0 P) z:0 hence a(C qi @p,)T = 0 hence a(C qi 8~~) = 0. This produces an isomorphism hom,( T, X) s ho&Q OS P, X) = hom,(Q, hom,(P, X)), whose composition with the natural map X -+ hom,(T, X) is just 4X . Therefore dx is bijective iff X -+ hom,(T, X) has this property, i.e., iff X E 5Z by the preceding proposition.
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